Majorana zero mode in the vortex of artificial topological superconductor

Superconductivity was discovered by Onnes,¹ which is a quantum phenomenon characterized by zero resistance and the Meissner effect below a certain temperature. Since then, a series of conventional s-wave superconductors (SCs) have been found in elements (e.g., lead), alloys (like Nb₃Ge), and compounds (such as niobium nitride²). The theory of these superconductors was developed by Bardeen, Cooper, and Schrieffer in 1957, which is called the BCS theory.³ The BCS theory attributes the superconductivity to the formation of Cooper pairs, which behave like bosons and can condensate at low temperatures. The quasiparticles in superconductors are protected by particle–hole symmetry in Bogoliubov–De Gennes (BdG) Hamiltonian. Apart from that, the unconventional superconductivity attracted explosive research interest in the past two decades. The paradigmatic example is the p-wave superconductivity, which manifests topological superconducting phases.^4–6 More importantly, it may host Majorana zero mode (MZM) that obeys exotic non-Abelian statistics and has potential applications for topological quantum computing.^7–9 

Topological superconductor (TSC) may be understood by mean-field BdG Hamiltonians. There are broadly two classifications of TSCs, intrinsic ones and artificially engineered ones. Intrinsic TSCs are those in which the pairing state emerges by virtue of the materials’ internal properties. It is worth noting that superconductors having spin-triplet pairing are, in most cases, topologically nontrivial. However, triplet pairing rarely exists in the nature. Sr₂RuO₄ was theoretically proposed to be the candidate of intrinsic TSCs. Nuclear magnetic resonance and muon spin rotation experiments showed several pieces of evidence of spin-triplet superconductivity with a definite chirality in Sr₂RuO₄.^10,11 Nevertheless, the realization of chiral p-wave superconductivity in Sr₂RuO₄ is still under debate due to the chiral domains having never been observed. Recently, UTe₂ showed a large intrinsic zero-temperature reservoir of gapless fermions that may also be an intrinsic TSC.¹² Due to the difficulty in searching for intrinsic TSCs from the few thousand superconducting compounds discovered so far, artificially engineered TSCs attracted significant attention in the past decade. Here, we mainly discuss the artificially engineered TSC in two-dimension, which is revolutionized by the discovery of a topological insulator.

The first topological non-trivial phase was found in the quantum Hall state,¹³ which is characterized by an integer number called Chern number.¹⁴ In general, Chern number is a prominent example of a topological invariant. To realize a quantum Hall system, one needs an extremely large magnetic field to break time-reversal symmetry explicitly. In 2005, quantum spin Hall state was proposed to exist in a two-dimensional topological insulator, which is characterized by a Z₂ topological invariant,^15,16 leading to the birth of a topological insulator.¹⁷ The quantum spin Hall state is protected by time-reversal symmetry and in which spin–orbit coupling plays an essential role. Soon afterward, HgTe/CdTe quantum wells were first experimentally demonstrated to be the quantum spin Hall topological insulator,^18,19 which owns a pair of edge states with a helical spin texture. The concept of topology was naturally extended from two-dimensional to three-dimensional systems, where there are four Z₂ invariants to fully characterize their topology.^20–22 Three-dimensional topological insulators are associated with a large bulk gap and a gapless surface state consisting of an odd number of Dirac cones. To date, the most well-known three-dimensional topological insulators are Bi₂Se₃ and its isostructural cousins Bi₂Te₃ and Sb₂Te_3.^23–27 The discovery of the topological insulator not only has had a dramatic impact on the field of topological materials^28–44 but has also paved the way for realizing two-dimensional TSC based on the unique surface state.

This is basically equivalent to the one for the two-dimensional chiral p-wave pairing superconductor. Hence, the vortex in such a superconductor also hosts MZM, similar to the two-dimensional p-wave superconductor. Besides, the new system is a time-reversal invariant because only conventional s-wave superconductivity is required in the proposal, which is a notable advantage compared with the intrinsic p-wave TSC that necessarily breaks time-reversal symmetry.

Following this approach, one can construct artificially engineered TSC experimentally. By fabricating heterostructures combined with three-dimensional topological insulators and s-wave superconductors, the two-dimensional TSC can be established as long as the superconductivity is induced into the topological surface states. To realize the superconducting proximity effect, a transparent interface of heterostructures is of the essence, and the choice of a suitable s-wave superconductor substrate is a challenging one to make. The construction of our artificial TSC system will be discussed in detail later.

The research of TSC has attracted extensive interest due to the relevance to MZM which is predicted to exist in the vortex. MZMs follow the Majorana equation that describes Majorana fermion⁴⁶ and are potentially capable of fault-tolerant topological quantum computation based on their non-Abelian statistics. However, evidence of MZM in the vortex of two-dimensional TSC is still a long-sought issue. We will briefly introduce three characteristic features of MZM in the following part.

Theoretically, a zero-bias peak (ZBP) observed in tunneling spectroscopy is the direct evidence for the existence of MZM. However, for most of the above superconductors, the superconducting gap Δ is about a few meV, while the Fermi energy is about several hundred meV, which leads to an interlevel distance $Δ 2/ E F$ of about only a few μeV. Such a small energy separation is below the resolution in actual experimental instruments. Consequently, the easily observed feature in the vortex is that a peak of density of states locates at zero energy with a symmetric shape.^49,50 The overlap of multiple CdGM states usually develops such a ZBP, which may emerge without MZM. The observation of ZBP alone is not enough to prove the existence of MZM. Alternatively, a two-dimensional TSC with a larger superconducting gap and smaller Fermi energy is helpful to distinguish MZM from CdGM states. In a way, more convincing verification is required to detect the MZM.

An MZM can be detected as a zero-bias conductance peak (ZBPC) in tunneling spectroscopy. Besides, the spatial distribution of such a quasiparticle excitation has been evidenced to be another intriguing feature.^51,52 Here, we mainly focus on the MZMs in the vortex cores of TSC, which are generated by proximity effect from an s-wave superconductor on topological surface states.

According to the previous work,⁵² the bulk bandgap between the valance and the conduction bands is taken as $2 ϵ 0$⁠, and Fermi level $μ$ is measured from the Dirac point of the topological surface states. In Fig. 2, the calculation presents the variation of the local density of state (LDOS) with energy and radius from the vortex center. One can see that when the ratio of $μ/ ϵ 0$ is relatively small, i.e., 0.5 and 1.0, MZM with zero energy distributes around the center of the vortex core [Figs. 2(a) and 2(b)], showing a Y shape of the LDOS. However, when the ratio of $μ/ ϵ 0$ is larger [Figs. 2(c) and 2(d)], the absence of MZM leads to a V-shape variation. In other words, the spectral distribution evolves from a V shape to a Y shape, indicating the existence of MZM to a degree. Moreover, the chemical potential plays a vital role in generating MZM, only when the Fermi level lies in the bandgap and SC gap is opened in the topological surface states.

The above-predicted characteristic demonstrates the emergence of MZM in a way, whereas the Fermi energy in different systems and the limited energy resolution of instruments somehow complicate these features, e.g., the mixture of MZM and CDGM states in a vortex cone. Fortunately, a spin-selective Andreev reflection (SSAR), which is a novel magnetic property, has been proposed to be a direct evidence of MZM.

SSAR was first pointed out in a TSC using a Rashba semiconducting wire in proximity to an s-wave superconductor where MZMs exist at two ends of the wire.^53,54 An incoming electron with the same spin of the MZM is reflected as a counterpropagating hole with the same spin [Fig. 3(a)], namely, an Andreev reflection, while an electron with opposite spin is always reflected as electrons with unchanged spin [Fig. 3(b)], resulting in a spin-polarized charge current.

The SSAR process can extend from a one-dimensional nanowire to a 2D TSC,^55,56 e.g., a topological insulator/superconductor heterostructure. While the MZM is not a spin eigenstate, at the center of the vortex core, the spin-wave function of the MZM is parallel to the magnetic field, and the local Andreev reflection of the MZM is spin selective, namely, it occurs only when the electron has the spin polarization parallel to the magnetic field [Fig. 3(c)]. Otherwise, an incoming spin-down electron will be reflected as a spin-down electron because of the mismatch of the spins of the incoming electron and the MZM [Fig. 3(d)].

Furthermore, SSAR is more likely to be detected in the 2D TSC system. For 1D nanowire systems, it always requires a large external magnetic field to generate MZMs, making it difficult to attribute the spin polarization dependence to the MZMs.

As discussed above, the realization of intrinsic TSCs that host MZMs is still a challenge up to now. Alternatively, an artificially engineered TSC, like the TI/SC heterostructure, is currently attracting significant attention. Molecular beam epitaxy (MBE) is the most important method to fabricate such a hybrid system.

MBE is an epitaxial process by which the growth of materials occurs under ultrahigh vacuum (UHV) conditions on a heated crystalline substrate by the interaction of adsorbed species supplied by atomic or molecular beams.⁵⁷ What counts is the precise control of thin film growth at the atomic scale. In the 1960s, the development of MBE facilitated the fabrication of heterojunction transistors and novel active devices with “tailor-made” characteristics and provided access to new device phenomena.

Therefore, the better the vacuum in our possession, the longer the mean free path we can get. In this sense, atomic or molecular beams will not interact with other impurities, which ensures the high purity and quality of the sample. Then, by heating highly purified solid sources with Knudsen effusion cells or electron beam evaporators, atomic or molecular beams are created.

Another key point is epitaxial growth. Atomic or molecular beams can only absorb, migrate, nucleate, react, and form high-quality films on suitable substrates. It is necessary for a substrate to share the same crystalline structure or a structure with a similar symmetry with the target material. Also, the lattice constant of the substrate should differ from that of the material by no more than 10%. The temperature of the substrate also plays a crucial part in the growth process. For simple substances, different growth modes may occur due to a variation in the temperature. Besides, the formation of compound films requires a certain temperature range.

MBE owns several features that result in its peculiar role in preparing epitaxial structures for both applications and fundamental studies. Such exciting features can be summarized as follows: (i) The UHV growth environment allows the preparation of high-purity and clean materials with fewer defects, like semiconductors with high electron mobilities. (ii) The growth is a non-equilibrium thermodynamic process because of the spatial separation of a solid source and a substrate, which will lead to a novel phase of material compared with an equilibrium process. (iii) The relatively small growth rates (in the order of 0.1 nm–1 nm/s or less) allow for the reliable growth of ML-thick layers. (iv) MBE can be equipped with diagnostic techniques, like reflection high-energy electron diffraction (RHEED) and a scanning tunneling microscope (STM). Based on in situ RHEED, the lattice structure, parameter, and growth rate of films can be obtained.^58,59 Moreover, an STM-MBE combined system is powerful for the structure determination of complex surfaces and for exploring novel phenomena in new quantum materials.⁶⁰ We will discuss its applications on artificially engineered TSC in the following.

The typical growth modes of MBE include the two-dimensional growth of films and the three-dimensional growth of islands. The former consists of step-flow mode and layer-by-layer mode or is called Frank–van der Merwe growth. The above two growth modes can form a flat terrace at the atomic scale. At a lower growth temperature, the films prefer to grow in the layer-by-layer mode, i.e., the subsequent layer will form on the previous one after it wholly forms. By increasing the growth temperature, the deposited adatoms will diffuse near the top area of steps on the substrate and transfer to the bottom, which results in the step-flow mode.

Three-dimensional growth includes Stranski–Krastanov and Volmer–Weber modes is more common in real life. The Stranski–Krastanov growth is a layer-plus-island mode. The deposited adatoms will form a monolayer first, then gather together and become islands. In contrast, the Volmer–Weber growth leads to the formation of islands on the substrate. The interaction strength between adatoms and the surface determines whether two-dimensional or three-dimensional growth will happen. If the adatom–adatom interactions are weaker than adatom–surface interaction, adatoms prefer to attach on the surface and form layers. On the contrary, it will result in a three-dimensional growth and form islands. What is noteworthy is that experimenters prefer a two-dimensional growth mode in order to obtain a flat surface and do the surface analysis, especially in an STM measurement. Beyond that, only in the layer-by-layer growth mode, we can achieve the intensity oscillation of RHEED and control the thickness of the film or the number of layers precisely.

The scanning tunneling microscope (STM) was invented by Gerd Binning and Heinrich Rohrer in 1981.^61,62 It has opened a new world of possibilities for revealing surface morphology and spatially resolved structure/electronic properties at the atomic level. The ultrahigh spatial resolution (∼0.1 nm in horizontal, 0.01 nm in vertical) of STM has contributed to scientific research in microcosm. More importantly, combined with extreme experimental conditions, e.g., ultrahigh vacuum, ultra-low temperature, strong magnetic field, STM itself is becoming increasingly powerful and is proved to be a versatile technique for many disciplines in condensed matter physics, chemistry, biology, and other areas.

The basic property of STM is quantum tunneling of electrons between two electrodes separated by a thin potential barrier (usually, a vacuum barrier). When the distance between tip and sample is in the order of 0.5–1 nm, and when a voltage is applied between them, it will lead to the formation of a normal metal/insulator/normal metal (NIN) junction. Meanwhile, electrons will tunnel regardless of the barrier and form a current that can be detected. Electrons with higher energy can pass through a barrier in quantum mechanics but prohibit in the classical scenario.

That is, during the STM measurement, the effective information we get is the tunneling current recorded between the tip and the sample. If we scan the sample, the variation in the tunneling current reflects the morphology of the sample at different positions in real space, and the image of the surface at the atomic scale is obtained. Besides, the $ρ s a m p l e(ε)$⁠, which is associated with the tunneling current, will be exhibited in some sense.

The tunneling conductance dI/dV is directly proportional to the DOS of the sample at a given energy (applied bias voltage). By measuring the dI/dV signal at different energies, we can obtain the scanning tunneling spectroscopy (STS), one of the fascinating potentials of STM. It allows us to measure the spatial and energy dependence of the sample DOS. Among most surface analysis technologies, STM is unique in resolving information both in real and reciprocal space with ultrahigh resolution. Also, unoccupied and occupied sample states can be measured with a positive and negative bias voltage, respectively.

Based on the STS measurement, another widely used technique is spectroscopic imaging, called dI/dV grid, which combines the atomic resolution in real space and in the sample DOS taken by STM. Usually, the topography image is obtained through scanning over the sample with a constant current by tip. During the measurement, the feedback is closed all the time to keep current and record the topographic information. A dI/dV grid is acquired in the process of a scan. When the tip moves to each point on the sample, the scan and feedback are interrupted to freeze the tip position and then measure the dI/dV signal at a single bias value or over an extended voltage range.

Then, the feedback is turned on, and the scanning is resumed. After the scan process, the topography image and a set of DOS maps at different energies are obtained simultaneously.

A dI/dV grid can directly measure the real-space distribution of the sample DOS both in occupied and in unoccupied states, which has been demonstrated to be a practical method in condensed matter physics. For instance, quasiparticle interference (QPI) that is acquired by the dI/dV grid has attracted increased attention in the study of the electronic properties of the unique states in quantum materials. The electron of the surface state in materials can be scattered due to surface imperfection, such as point and line defects, giving rise to a standing wave pattern, which can be achieved by the dI/dV grid. Following a Fourier transform, all allowed scattering vectors will be exhibited and open the possibility of energy dependence reciprocal-space spectroscopy. The method of QPI has been used to study a series of electronic systems including a high-temperature superconductor,⁶³ topological nontrivial phases as topological insulators, Weyl semimetals,^64–69 and nodal-line semimetals.^70,71

For the study of the superconductor, primarily type II superconductors, the dI/dV grid can be used to image the vortex when an external magnetic field is applied. The selected bias voltage for vortex imaging is usually at zero, because the amplitude difference of the sample DOS between the inside and the outside of the vortex is the largest at these energies. Therefore, it is necessary to use the dI/dV grid for the detection of MZM in the vortex core of TSC.

In condensed matter physics, the development of topological nontrivial phases pioneered a new understanding of phase transitions. TSC, the derivative of the topology concept in superconductivity, owns a superconducting gap in the energy-band spectrum to protect the occupied states that are generally analogous to insulators with topological property but different in energy scale. The realization of TSC actuates theoretical and experimental physicists working on the area due to the possibility of MZM existing in TSC. MZMs have attracted tremendous interest owing to their non-Abelian statistics and potential applications in topological quantum computation.

To date, several recipes have been proposed for realizing TSCs and detecting MZMs. Here, we briefly summarize predicted systems and experimental materials of the past decade in Table I. MZMs can emerge at the ends of 1D TSC or in the vortex core and edge of 2D TSC, as research on the former type TSC has mainly involved hybrid superconductor–semiconductor nanowire devices^72–75 and a chain of magnetic atoms on a superconductor.^76–78 As shown in Fig. 4(a), a semiconductor nanowire with strong spin–orbit interaction will split the twofold band into a spin-down and a spin-up band. In the presence of an external magnetic field B along the axis of the nanowire, i.e., a Zeeman field, a gap is opened at the band crossing point. When the Fermi energy is tuned inside the gap, the superconducting pairing will form between electron states of opposite momentum and spins by proximity the nanowire to a BCS superconductor, then combining this twofold degeneracy with an induced gap creates a TSC. The MZMs will arise at each end of the wire. Several pieces of evidence of MZMs have been observed in hybrid structures like the Al/InSb nanowire by tunneling measurements.^72–75 

Alternatively, one has proposed to use magnetic textures to emulate spin–orbit coupling and realize 1D TSC. For instance, based on transition metal chains with ferromagnetism, the large exchange interaction generates a fully occupied majority band along with states near the Fermi level due to a minority band [see Fig. 4(b)]. Suppose the chains are grown on a superconductor with strong spin-orbit coupling. In that case, the proximity-induced superconductivity will be topological due to the odd number of band crossings at the Fermi level. Such a proposed platform has been realized on self-assembled chains of Fe atoms formed on the surface of Pb(110), and edge-bounded MZMs have been observed by STM measurement.^76–78 

Apart from the 1D TSC we mentioned above, the research of 2D TSC attracted increased attention after a prediction made by Fu and Kane, who proposed an artificially engineered TSC based on an s-wave SC and the exotic properties of the electronic surface states of a TI. The proposal has been visualized in a TI/SC heterostructure like Bi₂Se₃/NbSe₂ and Bi₂Te₃/NbSe₂ systems by MBE.^56,79–82 Via a novel flip-chip technique, the helical pairing of the Dirac fermions was also realized in a Bi₂Se₃/Nb heterostructure,⁸³ and further analysis of surface evenness in the system is necessary for the research of MF. However, due to the limit of pairing potential of the conventional s-wave SCs, the proximity-induced pairing potential is relatively small for the above systems, which requires an extremely low temperature for the experiment. Naturally, unconventional SCs are supposed to construct the TI/SC heterostructure. Such an idea was initially put into action through growing Bi₂Se₃ films on a d-wave SC with a high superconducting transition temperature and formed Bi₂Se₃/Bi₂Sr₂CaCu₂O_8 + δ,⁸⁴ but the proximity-induced superconductivity was not demonstrated in this system.^85,86 This issue may root in the Fermi surface and lattice structural mismatch, strong electron correlation, and short superconducting coherence length. Alternatively, unconventional iron-based SCs were used as a substitute, and the absence of superconductivity is exhibited in FeSe/Bi₂Se_3.⁸⁷ Although superconductivity was observed in Bi₂Te₃/FeTe⁸⁸ and Bi₂Te₃/FeTe_0.55Se_0.45,⁸⁹ the question whether the induced superconductivity can be regarded as two-dimensional chiral p-wave pairing is still under debate. Besides, superconductors with topological surface states, like iron-based SCs, include FeTe_0.55Se_0.45^90–92 and (Li_0.84Fe_0.16)OHFeSe,⁹³ and other novel SCs include 2M-WS₂,⁹⁴ β-Bi₂Pd,^95,96 and PbTaSe₂⁹⁷ and can be described by the same model of TI/SC and they show the properties of TSC. It is worth noting that a gel of quantum vortices was observed in β-Bi₂Pd at low magnetic fields, which may allow to considerably simplify control over vortex positions and manipulation of MZM in the vortices.⁹⁸ The 2D topological superconductivity was realized in Pb/Co/Si(111) as well. By using a 2D Rashba superconductor coupled to a magnetic cluster, the propagating Majorana edge states and a single MZM in the vortex was revealed simultaneously.^99,100 Except for these platforms, a magnetic skyrmion of an even azimuthal winding number placed in proximity to an s-wave SC may host MZM.^101,102

Recently, a new type of TSC-superconducting topological crystalline insulator has been demonstrated experimentally.^103,104 Topological crystalline insulators (TCIs) are topologically nontrivial states of matter that the gapless surface states are protected by crystalline symmetry instead of time-reversal symmetry in TI. The typical TCIs have been verified in the SnTe-type IV–VI semiconductors. By fabricating atomically flat lateral and vertical Sn_1−xPb_xTe–Pb heterostructures [see Fig. 4(c)], one has investigated the proximity-induced superconductivity by STM, which suggests the presence of the topological superconductivity in superconducting Sn_1−xPb_xTe. More importantly, high-resolution scanning tunneling spectroscopy measurements have been performed on lateral Sn_1−xPb_xTe–Pb heterostructures via superconducting tips. The multiple in-gap states are resolved at 0.38 K. This work verified that the unique topological superconductivity of a TCI could be directly distinguished in the density of states. Therefore, the strong superconducting proximity effect of Sn_1−xPb_xTe–Pb heterostructures makes it a promising candidate for topological superconducting devices, and the detection of MZMs in such a system is in prospect.

With precise theoretical prediction and elaborate experimental realization, various significant TSCs have been discovered in different dimensions, which paves the way for a research of MZMs. In this review, we will introduce the experimental results of 2D TSCs, which are suitably investigated by STM because the MZMs exactly exists in the vortices of TSCs. Before this, the fabrication of the TI/SC heterostructure will be discussed in detail, which is a substantial achievement of an STM-MBE combined system.

Since the TI/SC heterostructure was predicted to be an artificial two-dimensional TSC, a series of works have been done to make the proposal come true. As one indispensable part of the heterostructure, the three-dimensional (3D) topological insulator is characterized by Z₂ topological invariants and possesses insulating gaps in the bulk and gapless states on surfaces. It has already been studied both in theory and in experiment.

Bi₂Te₃, along with the congeneric Bi₂Se₃ and Sb₂Te₃, has been demonstrated to be the simplest 3D topological insulator whose surface states consist of a single Dirac cone at the Г point. This class of stoichiometric materials is well known for their thermoelectric property. The bulk crystals of Bi₂Te₃ class of topological insulators are usually synthesized via a self-flux technique. Elemental starting materials in the fixed molar ratio are placed in alumina crucibles and sealed under vacuum in a quartz ampoule. The mixture of Bi and Te is heated to a high temperature and then slowly cooled down for a long period and form plate-like crystals with hexagonal edges. Generally, the growth method of bulk crystals will induce a mass of vacancies and anti-site defects, which result in a high carrier density, and the Fermi level will no longer locate in the bulk gap but in the conduction band, and, thus, the topological insulator is indeed conductive in the bulk. With the bulk electronic states at the Fermi level, it is difficult to characterize the pristine topological transport property and use them to develop topological devices that rely only on the behaviors of Dirac fermions. Alternatively, an intrinsic topological insulator can be obtained in the form of a thin film. Meanwhile, it is more precise to dope and engineer the band structure during the growth of a topological insulator thin film and more helpful for the construction of heterostructures and superlattices.

As we mentioned above, via molecular beam epitaxy, we can prepare high-quality thin films and precisely control the growth rate at the atomic scale. The Bi₂Te₃ class of a topological insulator, which possess a rhombohedral lattice with a space group of $R 3 ¯m$ (No. 166), is quite suitably synthesized by MBE growth.^105,106 As shown in Fig. 5(a), along the [111] crystallographic direction, the unit cell of Bi₂Te₃ contains five atomic layers with a stacking sequence of Te(1)–Bi–Te(2)–Bi–Te(1), forming a quintuple layer. It is noteworthy that Bi₂Te₃ is easily cleaved along the (001) surface due to the weak van der Waals interaction between adjacent quintuple layers. On the one hand, the layered characteristic of Bi₂Te₃ lowers the requirement of the symmetry and lattice constant matching between the thin film and the substrate. On the other hand, the layer-by-layer growth mode is helpful for controlling the thickness, which can be monitored by in situ RHEED during growth.

Inspired by the phase diagram of Bi–Te binary compounds, an ideal MBE growth of Bi₂Te₃ can be anticipated under a Te-rich condition with proper substrate temperature. On account of the different saturated vapor pressure of Bi and Te, the optimized substrate temperature should satisfy T_Bi> T_Sub≥ T_Te, where T_Bi and T_Te are the temperatures of Bi and Te Knudsen cells, and T_Sub represents the substrate temperature. The above temperature condition of the substrate avoids decomposing the film and helps to remove additional Te. Layer-by-layer molecular beam epitaxy growth of high-quality Bi₂Te₃ films has been achieved on the Si(111) substrate. In this systematic research, T_Bi, T_Sub, and T_Te are chosen to be 773, 543, and 543 K, respectively. Under optimal conditions, the Bi₂Te₃ film with 4 QL has been prepared on the Si(111)-7 × 7 surface. From the pattern measured by in situ RHEED, the sharp streaky pattern indicates that the film has an atomically flat surface morphology [see Fig. 5(b)], which lays the foundation of surface characterization. Accompanied by the real-time monitoring of RHEED, the intensity of the oscillation curves can be recorded [see Fig. 5(c)]. One period in the curve indicates that one QL of Bi₂Te₃ is formed because the diffraction is strong on the entire layer but weak on the half layer due to the existence of multiple edges. Thus, the accurate growth time of a single QL is obtained. Figure 5(d) shows a typical STM image of an 80-nm-thick film grown on Si(111), with a large terrace and edges being visible.

With MBE and in situ RHEED, the thickness of the Bi₂Te₃ thin film can be well controlled. Hence, it provides vast opportunities for physicists to study the novel phenomenon in two-dimensional limits. Beyond that, angle-resolved photoemission spectroscopy (ARPES) as an important surface analysis method that can directly measure band structures in momentum space is widely used in topological nontrivial phases.^107–124 The surface-sensitive character of ARPES leads to an extensive research of the surface state. The evolution of the surface state of the Bi₂Te₃ thin films with different thicknesses has been systematically studied by ARPES.¹⁰⁶ As shown in Fig. 5(e), there is a gap between the conduction band and the valence band for the 1 QL film. The topological surface states start to emerge when the thickness is equal or greater than 2 QL. Within the bulk gap, a Dirac-like band dispersion can be observed. In contrast, the thickness of Bi₂Se₃ that starts to recover the topological features is 6 QL, which is relatively larger than Bi₂Te₃. The lack of surface states in the 1 QL film is due to the strong coupling between top and bottom surfaces. Meanwhile, the inter-surface coupling becomes weaker as the film thickness increases. It is worth noting that with increasing thickness, the Fermi level gradually moves toward the bulk gap, which is a remarkable phenomenon, because we need to induce superconductivity into the surface state of the topological insulator to realize topological superconductivity. Therefore, to construct a two-dimensional TSC, the thickness limit of the topological insulator should remain in consideration.

To induce superconductivity into the surface state of the three-dimensional topological insulator, the first attempt is made by chemical doping. The superconductivity of Cu-doped Bi₂Se₃ has already been demonstrated,^125–132 and the additional requirement in such case is that the Cu atoms must randomly insert into the Bi₂Se₃ crystals at the space between adjacent Se–Bi–Se–Bi–Se quintuple layers. However, whether the zero-bias peak in the vortex of Cu_xBi₂Se₃ is a signature of MZM is still a controversial issue.¹³³ Hence, one needs to seek another approach to realize TSC.

Fortunately, if a topological insulator and a superconductor can be put together by the superconducting proximity effect, superconductivity can also be introduced into the surface state of a topological insulator. In this case, the TI/SC heterostructure needs to be prepared meticulously, because a sharp yet electronically transparent interface between two materials is indeed important to generate a strong proximity effect. The original suggestion of a TI/SC heterostructure by Fu and Kane is to grow SC on the top of the TI. However, due to the decomposition temperature limit of TI, most SCs cannot be grown on the surface of the TI or are unable to form an atomic flat terrace on which to do further measurement.

In 2012, we followed the opposite method and successfully prepared three-dimensional topological insulator thin films on the top of a conventional s-wave superconductor by MBE.⁷⁹ The superconductor substrate we chose was 2H-NbSe₂, a layered material with a superconducting transition temperature of around 7 K. The layered property of 2H-NbSe₂ overcomes the main challenge of lattice mismatch between the added layer and the host material. Meanwhile, atomically flat and large terraces of the 2H-NbSe₂ surface can be obtained by in situ cleavage. First, the well-known topological insulator Bi₂Se₃ film was grown on NbSe₂ (0001) by MBE under a similar condition compared with that grown on graphitized 6H-SiC (0001). A buffer layer, Bi (110) bilayer, was deposited on NbSe₂ in advance to ensure the flatness of the Bi₂Se₃ film¹³⁴ [see Fig. 6(a)]. Using in situ STM measurement, we show that the Bi₂Se₃ film with various thicknesses formed on the NbSe₂ substrate [see Figs. 6(b) and 6(c)], large and clean terraces, as well as the sharp edges demonstrate the high quality of our sample. As shown in Fig. 6(d), superconducting gaps are observed on the Bi₂Se₃ films with a thickness of 3 QL and 6 QL, which are featured with a dip at the Fermi level, and the coherence peaks are pronounced at 0.4 K due to the proximity effect of the NbSe₂ substrate. The results indicate that superconductivity is induced into the Bi₂Se₃ film due to the proximity effect of the NbSe₂ substrate.

The existence of topological surface states is confirmed by ARPES. The energy-band dispersion of the Bi₂Se₃ films on NbSe₂ is shown in Fig. 6(e). An energy gap is revealed when the thickness is 3 QL, and Rashba-type spin–orbital splitting of quantum well states is observed. The absence of Dirac cone surface states in 3 QL Bi₂Se₃ is attributed to the coupling between two opposite surfaces. As expected, when the film is thicker than 6 QL, surface states form the Dirac cone, and the Dirac point emerges at the binding energy of 0.45 eV. Together, these results provide important insights into the coexistence of superconductivity and a single helical Dirac cone in the Bi₂Se₃/NbSe₂ heterostructure, which resembles a spinless p_x+ ip_y-like superconductor.

The two-dimensional topological superconductivity of our system is further supported by spin- and momentum-resolved photoemission spectroscopy with sufficiently high resolution and at sufficiently low temperature.¹³⁵ A high-resolution ARPES dispersion map of a 4 QL Bi₂Se₃ film on NbSe₂ is shown in Fig. 6(f), and the Dirac surface states are revealed. The spin-resolved ARPES measurements are performed on the sample with the same thickness, as shown in Fig. 6(g), along the fixed momentum indicated by the dotted line in Fig. 6(f), and the spin polarization of the upper band is opposite to the lower band, which is in accord with the spin-momentum-locking nature of the topological surface state. The superconducting signals are visualized by symmetrized ARPES spectra with respect to the Fermi level at 1 K. Figure 6(h) shows the surface state Fermi surface of the 4 QL Bi₂Se₃ film on NbSe₂. By choosing five representative momentum-space cut-directions (indicated by the dotted lines), the helical surface state superconducting gaps are observed and shown in Fig. 6(i). This study proved that the superconductivity is induced into the surface states of Bi₂Se₃ by the proximity effect, which is quite significant, because the surface state and the bulk state of Bi₂Se₃ are coexisting at the Fermi level, and only the surface state of the topological insulator superconductivity can be regarded as two-dimensional topological superconductivity. Therefore, the next task is to exclude the conventional superconductivity from the bulk states, in other words, to replace the topological insulator with only Dirac surface states at the Fermi level and realize the desirable helical Cooper pairing.

Based on the above analysis, alternatively, our group then focused on the intrinsic topological insulator Bi₂Te₃ whose band structure only consists of a topological surface state at the Fermi level.⁸⁰ Under a Te-rich condition, we employed an elaborate growth method and prepared multiple QL Bi₂Te₃ thin films on top of NbSe₂ by MBE. The high quality of the Bi₂Te₃ film with a large terrace is characterized by STM [see Fig. 7(a)]. On 3 QL Bi₂Te₃ grown on NbSe₂, we measured the dI/dV grid at zero bias after applying a perpendicular magnetic field with 0.75 T at 0.4 K. The Abrikosov vortices with highly ordered hexagonal lattice are observed, which is the premise of the detection of MZM [see Fig. 7(b)]. The thickness of Bi₂Te₃ plays a crucial role in forming the topological surface state and further tuning the Fermi level into the energy range without bulk carriers. Figure 7(c) shows the evolution of the STS on Bi₂Te₃ films of various thicknesses grown on NbSe₂. The Dirac surface state arises at 3 QL, as the deformed U-shape segment emerges, which is similar to that of Bi₂Te₃ films grown on Si(111) substrates. The black arrows indicate the bulk conduction band minimum. With increasing film thickness, the conduction band minimum gradually shifts upward. When the thickness reaches 5 and 6 QL, the Fermi level is almost entirely in the bulk bandgap, which means if superconductivity is induced into this case, the Cooper pairing is only from the Dirac surface state, and the conventional superconductivity from other bands no longer exists. Besides, the superconducting gap is observed on Bi₂Te₃ films up to 11 QL grown on NbSe₂ at 0.4 K, and the gap value decreases exponentially as the thickness increases [see Figs. 7(d) and 7(e)], which is consistent with the behavior of proximity effect-induced superconductivity.

Therefore, our group has successfully fabricated a TI/SC heterostructure and ensured that superconductivity is induced into the topological surface state of TI via the proximity effect. More importantly, we proved that Bi₂Te₃/NbSe₂ is a more suitable platform than Bi₂Se₃/NbSe₂ to realize two-dimensional topological superconductivity. The existence of MZM in our configuration will be discussed in Sec. V.

MZMs are potentially capable building blocks of fault-tolerant topological quantum computation due to their non-Abelian statistics. Theoretically, the proximity effect from an s-wave superconductor on topological surface states would produce a two-dimensional TSC that can host MZM in vortices. In Sec. VI, we have discussed the fabrication of the artificial TSC and the demonstration of topological superconductivity in the TI/SC heterostructure. The next step is to exhibit the anticipant signature of MZM. As introduced in Sec. I, three characteristic features are closely related to MZM, which are ZBP, spatial distribution, and SSAR. In this section, we will mainly review our results about the detection of MZM in the vortex based on artificial two-dimensional spinless chiral p_x + ip_y wave TSC. Additionally, several compelling observations on ZBP in other platforms like iron-based superconductors will be discussed.

A ZBP in the vortex core of two-dimensional TSC may potentially correspond to MZM. Via ultra-low temperature STM, such a signal can be easily captured. By means of MBE, 5 QL Bi₂Te₃ thin films with a large terrace are prepared on NbSe₂, which is a suitable thickness to detect the MZM. After applying a small vertical magnetic field, we took the dI/dV grid on the sample at zero energy, and a single vortex with hexagonal shape is observed⁸⁰ [see Fig. 8(a)]. As shown in Fig. 8(b), a pronounced ZBP is found at the center of the vortex by STS measurement.⁸¹ What must be emphasized is that the ZBP in this TI/SC heterostructure is not only a result of MZM. Multiple CdGM states exist in the vortex, and the energy gap of these states is $0.83 Δ 2/ Δ 2 + E 2$⁠, where Δ is the superconducting gap and E is the energy difference between the Fermi level and the energy of Dirac point in this case. Considering that the superconducting gap size is Δ ∼ 1 meV and the Fermi level is E ∼ 100 meV in 5 QL Bi₂Te₃/NbSe₂, the energy separation between CdGM states and MZM is about 0.01 meV, which is out of the present energy resolution in our STM. Therefore, CdGM states and MZM together form a broad zero-energy peak in the quasiparticle excitation spectrum. To distinguish MZM from CdGM states, in other words, one has to increase the interlevel distance until it is comparable to the energy resolution. One way is to enhance the superconducting gap Δ, for which we need to find a superconductor with a higher superconducting transition temperature. Another way is to tune the Fermi level to the Dirac point as close as possible to reduce Fermi energy E.

Recently, iron-based superconductors have been proved to act as a new platform of TSC in which topological nontrivial bands and high-temperature superconductivity coexist.^90–93,136 Profiting from its large superconducting gap Δ and small Fermi energy E, MZM can be distinguished from the conventional CdGM states. In the vortex center of FeTe_0.55Se_0.45, ZBP and additional high-energy subgap features are observed in the dI/dV spectrum¹³⁶ [see the red curve in Fig. 8(c)]. When the spectrum is measured slightly off-center, these subgap features become prominent [see the blue curve in Fig. 8(c)]. The high-energy bound states originate from the CdGM states, and ZBP is well separated from the other CdGM states, which can be regarded as a persuasive evidence of the existence of MZM. However, this work shows that two distinct classes of vortexes are hosted in FeTe_0.55Se_0.45, one with ZBP, while the other only with half-odd-integer quantized CdGM states, which may complicate further topological quantum computation based on MZM. The separation of ZBP and CdGM states is also found on another iron-based superconductor (Li_0.84Fe_0.16)OHFeSe⁹³ [see Fig. 8(d)], and the CdGM states are more arresting in this case. For iron-based superconductors, the lower magnetic field H_c1 is relatively high, which means a large magnetic field is necessary to create superconducting vortices. The magnetic fields applied during the experiments on FeTe_0.55Se_0.45 and (Li_0.84Fe_0.16)OHFeSe are 6 T and 10 T, respectively, which are much larger than 0.1 T in our Bi₂Te₃/NbSe₂ system. Besides, topological superconductivity is also predicted in a transition metal dichalcogenide 2M-WS₂.⁹⁴ As shown in Fig. 8(e), a ZBP coexists with the splitting branches off the vortex center, and the splitting branches are attributed to CdGM states. Anisotropic MZMs originating from the anisotropy of the superconducting order parameter and Fermi velocity are discussed in this work.

Although it is generally tricky to establish the relationship between the apparent ZBP and MZM in our Bi₂Te₃/NbSe₂ system, the conductance contributed from MZM at zero energy may reveal MZM's existence. We carry out control experiments to demonstrate that the ZBP partly consists of MZM in the vortex of 5 QL Bi₂Te₃/NbSe_2.⁸¹ As shown in Fig. 9(a), ZBPs emerge at the vortex center on the surface of bare NbSe₂ due to the overlap of a series of CdGM states. The intensities of ZBPs gradually decrease when we increase the magnetic field from 0.025 to 1.0 T. The same behavior is observed on 2 QL Bi₂Te₃/NbSe₂ in which the Fermi level does not cut through the topological surface states [see Fig. 9(b)]. It is worth noting that the LDOS near a vortex core is virtually unchanged in a conventional s-wave superconductor. An abrupt change in the ZBP intensity is found on 5 QL Bi₂Te₃/NbSe₂ [see Fig. 9(c)], and the ZBP initially shows strong intensity at a field less than 0.1 T, but it suddenly decreases when the field is increased. The unusual change in ZBP intensity on 5 QL Bi₂Te₃/NbSe₂ is a direct result of the interaction between adjacent MZMs. For the type II superconductor, the vortex density is proportional to the magnetic field below the upper critical field. The typical vortex lattice has already been observed in our Bi₂Te₃/NbSe₂ sample with a thinner thickness [see Fig. 7(b)], which is consistent with those observed on the bare NbSe₂ surface. Due to the low vortex density at a small field, e.g., 0.025 T, the distance between vortexes is larger than the vortex size on the surface of 5 QL Bi₂Te₃/NbSe₂. So, each MZM in the vortex core is isolated, and the coupling between MZMs can be ignored, which will lead to a prominent ZBP dominated by MZM. However, as the density of vortexes increases with the magnetic field, the distance between the vortexes becomes smaller. When the field is increased to 0.18 T, the inter vortex distance is about 100 nm, which is comparable to the spatial distribution of MZM. Therefore, MZMs in the vortex cores will overlap and annihilate each other, resulting in a lower intensity of ZBP which only consists of trivial CdGM states. The abnormal behavior of ZBP potentially supports the existence of MZM in the vortex on 5 QL Bi₂Te₃/NbSe₂.

Except for the distinction of MZM from CdGM states in a ZBP, or the variation of ZBP intensity under different magnetic fields, the spatial distribution of MZM has been verified in a vortex of two-dimensional TSC. As discussed in Sec. I, when the ratio of Fermi energy to bulk bandwidth is small, MZM with zero energy will distribute around the center of the vortex core. We carried out a systematic measurement on the Bi₂Te₃/NbSe₂ heterostructure.⁸¹  Figure 10(a) shows an isolated vortex on the surface of 5 QL Bi₂Te₃/NbSe₂ taken by the dI/dV grid. A set of STSs is measured as a function of distance away from the vortex center in the 1 QL case [see Fig. 10(b)]. One observes a ZBP gradually splitting into two peaks as the distance increases. The same measurements are taken on Bi₂Te₃ with different thicknesses on NbSe₂ and shown in Figs. 10(c)–10(g) with false-color plot for a better view. The split peak positions are marked by red crosses, while two dotted lines are drawn to illustrate the splitting start points. For 1–3 QL Bi₂Te₃ thin films on NbSe₂, the peak splits just start from the vortex center, forming V-shape patterns [see Figs. 10(c)–10(e)], which is similar to previous works on conventional superconductors. Significant changes are taking place in the thicker Bi₂Te₃ films. As shown in Figs. 10(f)–10(g), in the sample with 4–6 QL Bi₂Te₃ films, the splitting points are away from the vortex center and further away from the center as the thickness increases, forming Y-shape patterns. From the summarized peak splitting location for samples with different thicknesses in Fig. 10(f), one can clearly find that the peak splitting positions move from the vortex center to a finite distance away from the center when the thickness of Bi₂Te₃ is larger than 3 QL.

To be specific, the variation of peak splitting positions is an inevitable consequence of the formation of MZM. From previous ARPES work, the topological surface states are not formed on the 1 QL Bi₂Te₃ film. In addition, the DOS at the Fermi level is dominated by bulk carriers on 2 and 3 QL Bi₂Te₃ films, although surface bands have formed. Hence, the superconducting Cooper pairing is from bulk states on 1–3 QL Bi₂Te₃ grown on NbSe₂, which is similar to conventional superconductors, and MZM should not appear. This situation changes in the thicker samples. The proximity effect-induced superconductivity is dominated by Dirac surface states on 4 QL Bi₂Te₃ and entirely exists in the helical Cooper pairing channel on five and six cases. In these circumstances, MZM forms in the vortex and whose spatial distribution is in the range of ∼40 nm away from the vortex center, according to the theoretical work on a Nb/Bi₂Se₃/Nb sandwich structure.⁵¹ Therefore, MZM in the range enhances the zero-bias LDOS and leads to the variation of peak splitting positions. Compared with the distribution of CdGM states, our results demonstrate the existence of MZM in the center area of the vortex on Bi₂Te₃/NbSe₂ with a suitable thickness of Bi₂Te₃. To guarantee that our system is in the intrinsic regime, in other words, only electrons of surface states participate in the superconducting pairing and become two-dimensional spinless chiral p_x + ip_y wave TSC. We choose 5 QL Bi₂Te₃/NbSe₂ as the platform for unearthing more evidence for MZM inside the vortex core.

Our group conducted the experiment via a spin-polarized scanning tunneling microscopy.⁵⁶ First of all, a magnetic tip is necessary to control the spin of the electron, and its coercive field should be larger than the applied field during the measurement. Otherwise, the spin of the incoming electron is no longer fixed. After careful consideration, W tips coated with ferromagnetic Fe are used to probe vortex core states. After transferring Fe/W tip to STM, we applied a magnetic field of 2 T to magnetize the tip and slowly reduced the field back to zero. The spin orientation of tips can be tuned by the direction of the magnetic field.

Based on the previous work, MZM exists in the vortex of 5 QL Bi₂Te₃/NbSe₂, so we carry on the experiment on the same platform. The sample temperature in the measurement is 30 mK obtained by dilution refrigerator, if not specified. As shown in Fig. 11(a), a vortex is observed on the sample under a magnetic field of 0.1 T, which helps us to locate the geometrical center of a vortex. Then, STS measurements are taken under various spin orientations both for incident electron from tips and for MZM in the vortexes. The out-of-plane field directions applied to the samples are shown by B↑ or B↓, while the spin polarizations of tips are shown by M↑ or M↓. We measure the STS at a position 10 nm away from the center, and the zero-bias conductance is exactly the same under two configurations indicated in the figure [see Fig. 11(b)]. As expected, the results change dramatically as we measure STS in the vortex center where MZM exists. By keeping the spin of MZM as fixed (B↑), we use a tip with two spin polarizations to measure STS in the vortex center, and it is revealed that the zero-bias conductance in the parallel field-tip configuration is higher than that in the antiparallel configuration [see the left two subpanels in Fig. 11(c)]. The same phenomenon is observed even we reverse the external magnetic field, namely, reverse the spin of MZM [see the right two subpanels in Fig. 11(c)]. The above data clearly exhibit the SSAR of MZM in which the parallel spin configuration enhances the zero bias. Besides, our theoretical calculations fit quite well with the experimental results [see Figs. 11(d) and 11(e)].

To further support the observation on 5 QL Bi₂Te₃/NbSe₂, we take the same measurement on samples without MZM. As shown in Figs. 12(a) and 12(b), on 3 QL Bi₂Te₃/NbSe₂ or bare NbSe₂ in which no MZM exists in the vortexes, the zero-bias conductance shows no obvious difference either in the parallel field-tip configuration or in the antiparallel configuration. In addition, the coupling of adjacent MZMs will enhance under a larger magnetic field due to the density of vortex increase. Therefore, we apply a magnetic field of 0.22 T to 5 QL Bi₂Te₃/NbSe₂ in which MZM should be destroyed by intervortex coupling, and the signal of SSAR also disappears as expected [see Fig. 12(c)]. Consequently, the SSAR process can only be accounted for the spin property of MZM, which provides direct evidence of MZM in artificial two-dimensional TSC.

The detection of MZM is significant both in fundamental physics and applications in topological quantum computation. Intrinsic p-wave TSC that may host MZM remains elusive. Alternatively, artificial two-dimensional TSC proposed by Fu and Kane has been sought in the past decade. As a result of the superconducting proximity effect from a conventional superconductor, the induced superconductivity in the surface states of a three-dimensional topological insulator would resembled a two-dimensional p-wave TSC. Combining the in situ MBE and STM techniques, we have fabricated high-quality topological insulator thin films on s-wave superconductor NbSe₂ and revealed the coexistence of a nontrivial topology and superconductivity in Bi₂Se₃/NbSe₂ and Bi₂Te₃/NbSe₂ heterostructures. The existence of MZM is proved in the vortex of 5 QL Bi₂Te₃/NbSe₂. The spatial distribution of ZBP is observed on 5 QL Bi₂Te₃/NbSe₂, which provides evidence for the existence of MZM. Besides, MZM is distinguished from trivial CdGM states in other platforms. More importantly, the SSAR process generated by the spin property of MZM is captured at the center of the vortex core on 5 QL Bi₂Te₃/NbSe₂, which demonstrates the discovery of MZM indisputably.

Our experimental achievements offer an ideal platform for fault-tolerant quantum computation based on MZM, and much effort is needed in the near future. To realize the braiding operations of MZM, we need to control the individual vortices and manipulate the MZM. One way is to use the STM tip to move the vortex. The local manipulation of single vortices by low-temperature magnetic force microscopy in a thin film of superconducting Nb has been reported.¹³⁷ In our STM system, a tip made of magnetic materials will be used. By reducing the distance of tip and vortex, the vortex and tip will be coupled. Then we move the tip, and the vortex will move simultaneously. Another way is to build arrays of electrically controllable pining centers beneath the Bi₂Te₃/NbSe₂ heterostructure.¹³⁸ By applying a voltage to the arrays (e.g., the piezoelectric material layer, the magnetoelectrical layer, and the conducting layer) to trigger the pining interactions and suppress superconductivity locally, the vortices with MZM are electrically controllable, giving rise to a much faster braiding rate and easy manipulation.

Among the various TSC systems, our TI/SC heterostructure possesses several technical advantages. The multiple-layer structure and the film preparation method, namely, MBE, all reflect the capability of the modern semiconductor industry. For instance, the product line of a GaAs/GaAlAs heterostructure, which is the building block of optoelectronic devices, can be immediately changed to grow a TI thin film with almost no modification of the facility. The microfabrication method, like lithograph and etching, may also be applied to build TI/SC based quantum devices with the aim to realize braiding operations. In summary, we believe that the TI/SC based TSC is not only a significant material in fundamental research but also a feasible system in semiconductor modern industry.

We thank the Ministry of Science and Technology of China (Grant Nos. 2019YFA0308600, 2020YFA0309000, 2016YFA0301003, and 2016YFA0300403), NSFC (Grant Nos. 11521404, 11634009, 92065201, 11874256, 11874258, 12074247, 11790313, and 11861161003), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB28000000), and the Science and Technology Commission of Shanghai Municipality (Grant Nos. 2019SHZDZX01, 19JC1412701, and 20QA1405100) for partial support.

The data that support the findings of this study are available within the article.